Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
322 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | D3×D9 |
| commutator, hypocenter | C3×C9 | C22×C16 |
| Frattini | C3×C8 | C2×D32 |
| Fitting | C2×C3×C8×C9 | C22 |
| socle | C22×C32 | D3×C8 |
| prime radical | C22×C3×D9 | C8 |
| cosocle | C3×C8×C9 | C23 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊(C9⋊C16) | C2 | semidirect |
| 3. C2×C9×C3⋊C16 | C2 | semidirect |
| 4. D9×C3⋊C16 | C2 | direct (N × H) |
| 12. C3⋊(C9⋊C16) | C22 | semidirect |
| 13. C9×C3⋊C16 | C22 | semidirect |
| 50. C2×C3×D9 | C16 | semidirect |
| 62. C2×C3⋊C16 | D9 | direct (N × H) |
| 84. C2×C3×C9 | C2×C16 | semidirect |
| 86. C3×D9 | C2×C16 | semidirect |
| 102. C3⋊C16 | C2×D9 | direct (N × H) |
| 111. C2×D9 | C3⋊C16 | direct (N × H) |
| 122. C3×C9 | C22×C16 | semidirect |
| 147. C2×C9 | C2×C3⋊C16 | semidirect |
| 149. D9 | C2×C3⋊C16 | direct (N × H) |
| 174. C9 | C22×C3⋊C16 | semidirect |
| 186. C2×C3 | C16×D9 | semidirect |
| 198. C3 | C2×C16×D9 | semidirect |
| 201. C2 | D9×C3⋊C16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C8×D9 | 2 |
| C2×C3⋊(C9⋊C16) | 2 |
| C2×C9×C3⋊C16 | 2 |
| D9×C3⋊C16 | 2 |
| C2×D3×C3⋊C16 | 3 |
| C2×C16×D9 | 3 |